MathDB

a) Given a circle with two inscribed triangles

T_1

and

T_2

. The vertices of

T_1

are the midpoints of the arcs with the ends in the vertices of

T_2

. Consider a hexagon -- the intersection of

T_1

and

T_2

. Prove that its main diagonals are parallel to

T_1

sides and are intersecting in one point.
b) The segment, that connects the midpoints of the arcs

AB

and

AC

of the circle circumscribed around the

ABC

triangle, intersects

[AB]

and

[AC]

sides in

D

and

K

points. Prove that the points

A,D,K

and

O

-- the centre of the circle -- are the vertices of a diamond.

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